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Chapter 14
Inductive Transients
Source: Circuit Analysis: Theory and Practice Delmar Cengage Learning
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Transients
Voltages and currents during a transitional interval
Referred to as transient behavior of the circuit
Capacitive circuit
Voltages and currents undergo transitional phase
Capacitor charges and discharges
Inductive circuit
Transitional phase occurs as the magnetic field builds and collapses
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Voltage Across an Inductor
Induced voltage across an inductor is proportional to rate of change of current
If inductor current could change instantaneously
Its rate of change would be infinite
Would cause infinite voltage
Infinite voltage is not possible
Inductor current cannot change instantaneously
It cannot jump from one value to another, but must be continuous at all times
t
iLv L
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Transient due to Inductor
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Inductor Voltage
Immediately after closing the switch on an RL circuit Current is zero
Voltage across the resistor is zero
Voltage across resistor is zero Voltage across inductor is source voltage
Inductor voltage will then exponentially decay to zero
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Open-Circuit Equivalent
After switch is closed (t=0+)
Inductor has voltage across it and no current through it
Inductor with zero initial current looks like an open circuit at instant of switching
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Initial Condition Circuits
Voltages and currents in circuits immediately after switching (t=0+) Determined from the open-circuit equivalent
By replacing inductors with opens We get initial condition circuit
Initial condition networks Yield voltages and currents only at switching
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Circuit Current and Circuit Voltages
Current i(t) in an RL circuit is an exponentially increasing function of time
Current begins at zero and rises to a maximum value
Voltage across resistor VR is an exponentially increasing function of time
Voltage across inductor VL is an exponentially decreasing function of time
L
Rt
R eEv 1
LRt
L eEv /
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Time Constant
= L/R , units are seconds
The larger the inductance
The longer the transient
The larger the resistance
The shorter the transient
As R increases
Circuit looks more and more resistive
If R is much greater than L, the circuit looks purely resistive
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Example: RL Transients
Given E=50V, R=10, and L=2H, determine i(t).
or = L/R=0.2s
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Example: RL Transients
Given E=50V, R=10, and L=2H, determine i(t).
= L/R=0.2s
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Interrupting Current in an Inductive Circuit
When switch opens in an RL circuit
Energy is released in a short time
This may create a large voltage
Induced voltage is called an inductive kick
Opening of inductive circuit may cause voltage spikes of thousands of volts
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Interrupting a Circuit
Switch flashovers are generally undesirable
They can be controlled with proper engineering design
These large voltages can be useful
Such as in automotive ignition systems
It is not possible to completely analyze such a circuit
Resistance across the arc changes as the switch opens
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Example: Interrupting a Circuit
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Inductor Equivalent at Switching
Current through an inductor
Same after switching as before switching
An inductance with an initial current
Looks like a current source at instant of switching
Its value is value of current at switching
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De-energizing Transients
If an inductor has an initial current I0, equation for current becomes
' = L/R. R equals total resistance in discharge path
Voltage across inductor goes to zero as circuit de-energizes
Voltage across any resistor is product of current and that resistor. Voltage across each of resistors goes to zero
T
where
RIV
eVv '/t
L
00
0
'/teIi 0
'/ τt
R eIRv 0
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De-energizing Transients
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More Complex Circuits
For complex circuits
Determine Thévenin equivalent circuit using inductor as the load
RTh is used to determine time constant
= L/RTh
ETh is used as source voltage
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Example1: More Complex Circuits
Switch is closed
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Example1: More Complex Circuits
Switch is closed (Cont’d)
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Example1: More Complex Circuits
Switch is opened
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Example1: More Complex Circuits
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Example2: More Complex Circuits
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Transient Analysis Using Computers
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Transient Analysis Using Computers
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Problem: Determine iL and vL
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Problem: Determine iL and vL